Ординатура / Офтальмология / Английские материалы / Optics of the Human Eye_Atchison, Smith_2000

Visual axis and achromatic axis: angle psi (1/1)

Thibos et al. (1990) estimated this angle from the equation

where d is the distance between the visual axis and line of sight at the entrance pupil and EN is the estimate of the distance between the

entrance pupil and the nodal point (Figure 4.2). Based on either the Gullstrand No. 1 or

No. 2 eyes, the distance EN is 4.0 mm. An

approximate, but sufficiently accurate, method for determining d was given earlier in

this chapter. Using five subjects, Thibos and co-workers determined a range of angles from -1.2° to +5.3°, with a mean of +2.1° (positive angles indicate the visual axis is inclined nasally to the achromatic axis in object space).

Fixation axis and optical axis: angle gamma (}1

Figure 4.8 shows the relationship between angles rand a. In this figure, y is the distance

between the optical axis and the fixation target T, N is the front nodal point, C is the

centre-of-rotation of the eye, and w is the distance from the projection of T, onto the

optical axis, to the cornea at V. We have the equations

Angle r is within 1 per cent of angle a for object distances greater than 50 em.

.Y

TFixation target

Figure 4.8. Determination of the angle Yo

Axes of tileeye 37

Summary of main symbols

Ccentre-of-rotation of the eye

Tfixation target

Vintersection of the optical axis with the

cornea

d distance between visual axis and line of

sight at the entrance pupil

a angle between visual axis and optical axis

rangle between fixation axis and optical axis

Aangle between pupillary axis and line of sight

'" angle between visual axis and achromatic axis

References

Alpern, M. (1969). Specification of the direction of regard. In Muscular Mechanisms, Ch. 2. Vol.3 of TheEye, 2nd edn (H. Davson, ed.), pp. 5-12. Academic Press.

Cline, D., Hofstetter, H. W. and Griffin, J. R. (1989).

Dictionary of Visual Science, 4th edn. Chilton Trade Book Publishing.

Franceschetti, A. T. and Burian, H. M. (1971). L'angle kappa. Bull. Mem. Soc. Fr. Ophtalmol., 84, 209-14.

Grosvenor, T. P. (1996). Primary Care Optometry, 3rd edn, Ch. 6. Butterworth-Heinemann.

Ivanoff, A. (1953). Lesaberrations de l'oeil. Leur role dans

I'accommodation. Editions de la Revue d'Optique

Theorique et lnstrumentale. Masson and Cie.

Le Grand, Y. and EIHage, S. G. (1980). Physiological Optics

(translation and update of Le Grand, Y. (1968). La dioptrique de I'oeil et sa correction. In Optique Physiologique, vol. 1), pp. 71-4. Springer-Verlag.

Loper, L. R. (1959). The relationship between angle lambda and the residual astigmatism of the eye. Am. J.

Optom. Arch.Am. Acad. Optom., 36, 365-77.

Maloney, R. K. (1990). Corneal topography and optical

zone location in photorefractive keratectomy. Refract. Corneal SlIrg., 6, 363-71.

Mandell, R. B. (1994). Apparent pupil displacement in videokeratography. CLAD t: 20, 123-7.

Mandell, R. B. (1995). Location of the corneal sighting

38 Basic optical structure of the humall eye

centre in videokeratographyI. Refract. (ameal Surg.,11,

253-8.

Mandell, R. B. and Horner, D. (1995). Alignment of videokeratographs. In (amen/ Topograplry: Tire State of tire Art (J. P. Gills, D. R. Sanders, S. P. Thornton et al.

eds.), Ch. 2. Slack Incorporated.

Mandell, R. B., Chiang, C. S. and Klein, S. A. (1995). Location of the major corneal reference points. Optom.

Vis. Sci., 72,776-84.

Mandell, R. B. and St Helen, R. (1969). Position and curvature of the corneal apex. Am. I. Optom. Arch. Am. Acad. Optom., 46, 25-9.

Martin, F.E. (1942). The importance and measurement of angle alpha. Br. J. Ophtlral., 3, 27-45.

Millodot, M. (1993). Dictionary of Optometry, 3'd edn,

Butterworth-Heinemann.

Rabbetts, R. B. (1998). Bennett and Rabbetts' Clinical Visual Optics, 3,d edn., Ch.12. Butterworths.

Simonet, P. and Campbell, M. C. W. (1990). The optical transverse chromatic aberration of the fovea of the human eye. Vision Res., 30, 187-206.

Thibos, L. N., Bradley, A., Still, D. L. et al. (1990). Theory and measurement of ocular chromatic aberration.

Visioll Res., 30, 33-49.

Tscherning, M. (1990). Physiologic Optics, pI edn. (translated from the original French edition by C. Weiland). The Keystone.

Introduction

We can construct model eyes using population mean values for relevant ocular

parameters. This can be done at different levels of sophistication. If we assume that the

refractive surfaces are spherical and centred on a common optical axis, and that the refractive indices are constant within each medium, this gives a simple family of models

referred to as paraxial schematic eyes. Paraxial schematic eyes are only accurate

within the paraxial region. They do not accurately predict aberrations and retinal

image formation for large pupils or for angles at more than a few degrees from the optical axis. The paraxial region is defined in geometrical optics as the region in which the replacement of sines of angles by the angles leads to no appreciable error. If we limit the errors to less than 0.01 per cent, this limits object field angles to less than 2° and the entrance pupil diameter to less than 0.5 mm.

Paraxial schematic eyes serve as a framework for examining a range of optical

properties. The location of the paraxial image plane or calculation of the paraxial image height has many useful applications. Information can be obtained from schematic eyes concerning magnification, retinal illumina-

tion, surface reflections (e.g. Purkinje images), entrance and exit pupils, and effects of refractive errors. A study of cardinal points of

the systems can also have practical applications, such as the observation that the

second nodal point moves little on accom-

modation and therefore that angular resolution is expected to change little with

accommodation. Further applications to retinal image formation are discussed in Chapters 6 and 9.

For accurate determinations of quantities such as large retinal image sizes and image

quality due to aberrations, we need more realistic models than the paraxial schematic eyes. These are referred to as finite or wide angle schematic eyes. These include one or

more of the following features: non-spherical refractive surfaces, a lack of surface alignment along a common axis, and a lens gradient refractive index.

Historically, paraxial schematic eyes have had uniform refractive indices, and it might be considered that schematic eyes with gradient indices must be finite eyes because the gradient index influences aberrations. However, replacing a uniform refractive index

by a gradient index affects the paths of

paraxial rays and hence paraxial properties, and thus gradient indices may be included in paraxial model eyes.

This chapter considers paraxial schematic eyes only. A discussion of finite model eyes is given in Chapter 16, following a review of the monochromatic aberrations of real eyes in Chapter 15.

40 Basic optical structure of tilellllll/all "y"

Development of paraxial schematic eyes

The historical development of the under-

standing of the optical system of the human eye has been described in detail by Polyak (1957). The lens was believed to be the

receptive element of the eye for 13 centuries following the work of Galen in 200 AD.

Leonardo DaVinci (c. 1500 AD) proposed that the lens is only one element of the refractive

system which forms a real image on the retina. In 1604, Kepler realized that the image is

inverted; this was verified by Scheiner 15

years later. The first clear, accurate description of the eye's optical system was given by Descartes in 1637 in his La Dioptrique, which also included the first publication of what has become known as Snell's law of refraction.

The first physical model of the eye was probably that of Christian Huygens (1629-95).

Smith (1738) described Huygens's eye as consisting of two hemispheres representing the cornea and retina respectively, with the

retinal hemisphere having a radius of curvature three times that of the corneal

hemisphere. The two hemispheres were filled

with water and a diaphragm was placed between them.

Young (1801) discussed the optics of the eye and presented data, some of which are close to present day values. He gave the anterior corneal radius as 7.9 mm, and the anterior and posterior lenticular radii of curvature as 7.6 mm and 5.6 mm respectively. The anterior

chamber depth was given as 3.0 mm. His refractive index for the aqueous and vitreous media was 1.333 (water), and that for the lens

was 1.44.

According to Le Grand and EI Hage (1980), Moser, in 1844, was the first to construct a

schematic eye, but this was hypermetropic because it had a very low value for the refractive index of the lens. The first 'accurate'

schematic eye has been attributed to Listing. In 1851, he described a three refracting

surfaces schematic eye with a single surface cornea and a homogeneous lens, with an aperture stop 0.5 mm in front of the lens. Helmholtz (1909, p. 152) modified Listing's schematic eye by changing the positions of the lenticular surfaces. He also gave this model in a form accommodated to a distance of 130.1

mm in front of the corneal vertex. Helmholtz

(1909, pp. 95-96) also described a much

simpler schematic eye designed by Listing. This contains only one refracting surface (the cornea), and is called a reduced eye.

Tscherning (1900) published a four refracting surfaces schematic eye containing a posterior corneal surface, which he claimed to be the first to measure.

Gullstrand (1909) used a comprehensive analysis of ocular data to construct a six

refracting surface schematic eye that used a four surface lens with the lenticular complexity aimed at accounting for refractive

index variation within the lens. This schematic eye is referred to as Gullstrand's

number 1 (exact) eye. Gullstrand presented this eye at two levels of accommodation. Gullstrand also presented a simplified version referred to as Gullstrand's number 2 (simplified) eye, also at two levels of accommo-

dation. This simplified eye contains three

refracting surfaces, with only one corneal surface and a zero lens thickness.

Emsley (1952) presented a modified version

of Gullstrand's simplified eye. Emsley gave the lens the thickness that it has in

Gullstrand's exact eye, and changed the

aqueous, vitreous and lens refractive indices. This modified eye is sometimes called the

Gullstrand-Emsley eye. Emsley also presented a reduced schematic eye.

As well as the Gullstrand exact eye, the Gullstrand-Emsley eye and Emsley's reduced eye, another popular schematic eye is Le Grand's 1945 four refracting surfaces eye, which is referred to as Le Grand's full theoretical eye (Le Grand and El Hage, 1980). It is a modification of Tscherning's schematic

eye. Le Grand also presented a simplified three refracting surfaces model with a single corneal surface and a lens of zero thickness. The lack of lens thickness limits the usefulness of this particular model.

More recently, Bennett and Rabbetts (1988, 1989) presented a modification of the Gullstrand-Emsley eye, which they justified

on the grounds that the data used to construct the earlier eye was from a restricted number of eyes and that the mean power is closer to 60 D than previously thought. They used the data from the study of Sorsby et al. (1957), which was based upon 341 eyes (mostly pairs of left and right eyes) with mean equivalent power of 60.12 ± 2.22 D.

Other schematic eyes have been proposed from time to time. For example, Swaine (1921)

gave details of several eyes referred to as Matthiessen B, D and G eyes, and Laurance I

and II eyes.

Blaker (1980) described an adaptive

schematic eye. It is a modified Gullstrand number 1 paraxial schematic eye, in which the lens has been reduced to two surfaces but is given a gradient refractive index. The lens gradient index, lens surface curvatures, lens thickness and the anterior chamber depth vary as linear functions of accommodation.

Blaker (1991) revised his model to include aging effects, with the lens curvatures, lens

thickness and anterior chamber depth altering in the unaccommodated state as a function of

age.

Some of the above mentioned eyes are discussed in greater detail later in this chapter, and constructional details of some eyes are

given in Appendix 3.

Gaussian properties and cardinal points

One of the main applications of paraxial

schematic eyes is predicting the Gaussian properties of real eyes. Of these, probably the

most important are the equivalent power F, positions of the six cardinal points (F, F', P, P',

Nand N') and the positions and magnifications of the pupils. We can use the paraxial

optics theory described in Appendix 1 to determine these properties. The Gaussian properties are given for specific schematic

eyes in Appendix 3, and Table 5.1 shows a limited amount of data.

Equivalent power and cardinal points

The cardinal points are defined in Chapter 1 (Cardinal points). Figure 1.1 shows nominal

positions of these in the emmetropic relaxed eye.

There are a number of useful equations connecting the cardinal points, including:

where nand n' are the refractive indices of

object space (air) and image space (the vitreous) respectively.

42 Basic optical structureo( tI,,'11I11I11I1I eye

Approximate mean values

Since the mean equivalent power of the eye is close to 60 D and the values of /1 and /1' are 1.0 and 1.336 respectively, we can calculate expected approximate mean values of the above quantities. These are:

F=60D

FP = N'F' = 16.67mm

P'F' = FN = 22.27 mm

PN =P'N' = 5.6 mm.

The aperture stop and entrance and exit pupils

After the equivalent power and positions of the cardinal points, probably the next most important Gaussian properties of an eye are the aperture stop and pupil formation. The aperture stop of an eye is its iris. Reduced eyes do not have an iris, but we can place an aperture stop in the plane of the cornea or at some other suitable position. The image of the aperture stop formed in object space, that is, the image of the iris as seen through the cornea, is called the entrance pupil. The image of the aperture formed in image space is

called the exit pupil. These concepts are discussed fully in Chapter 3.

Position and magnification of entrance pupil

For schematic eyes with a single surface cornea, the calculations are simple. In this case, we can use the lens equation given in Appendix 1,

Figure 5.1 shows the path of a paraxial ray that can be used to locate the image of the iris. I is the anterior chamber depth, I' is the apparent anterior chamber depth, /1 is the

and the pupil magnification MEA' defined as the ratio of the entrance pupil diameter to that

Figure 5.1. The formation of the entrance pupil of the

eye and its relationship to the iris in a schematic eye with a single surface cornea.

of the stop, is given by

The standard sign convention was used in the development of these equations, with distances to the left of the refracting surface being negative and distances to the right being positive. Distances I and l' are negative, although usually we express the final answers in a positive form.

Example 5.1: Calculate the position and magnification of the entrance pupil of the

Gullstrand-Emsley simplified relaxed eye.

Solution: From the Gullstrand-Emsley

schematic eye data given in Appendix 3, we have

/I = 4/3

/I' =1

I = -3.6 mm and F = 42.735 D.

Substituting these data into equations (5.5) and (5.6) gives

and

NT =(4/3) x (-3.052) = 1.1304 fA 1 x (-3.6)

Thus the entrance pupil is 3.05 mm inside the eye, compared with a distance of 3.6 mm for the actual pupil. The entrance

pupil is also 13 per cent larger than the

actual pupil. The pupil position is shown in Table 5.1, along with the values for

other schematic eyes.

Paraxial marginal ray and paraxial pupil ray

These are two special paraxial rays introduced and defined in Chapter 3 (Entrance and exit

pupils). As can be seen from Figure 3.3, the

paths of these rays depend upon the position of the object/image conjugates, field size and

the position of the aperture stop and its diameter. Here, as a rule, we denote the

marginal ray angles and heights by the respective symbols u and h and the paraxial

pupil ray angles and heights by the respective symbols Il and 11. The details of these rays (angles and heights) are given by Smith and

Atchison (1997) for some schematic eyes with an entrance pupil diameter of 8 mm and a

field-of-view of angular radius 5°.

Paraxial pupil ray angle ratio m

A quantity that is useful in the calculation of retinal image sizes is the ratio m of the

paraxial pupil ray angles

The angles Il and Il' are the angles of the

paraxial pupil ray in object and image space respectively, as shown in Figure 5.2. They are

related by the paraxial refraction equation (Appendix 1)

where F is the equivalent power of the eye and 11 is the ray height at the principal planes. Equation (5.8) can be transposed to give

where n has a value of 1 for air.

From Figure 5.2, within the paraxial approximation we have

which shows that the value of m depends

upon the refractive index of the vitreous which is fixed, the distance of the entrance pupil Tfrom the front principal point and the

equivalent power F. The values of both Tand F depend upon accommodation level. For a

typical schematic eye, T "" 1.5 mm, F:::: 60 D and n' ::: 1.336, giving m "" 0.82. Precise values for particular schematic eyes and at different

levels of accommodation are given in Table 5.1. Equation (5.11) can be manipulated into the following form

where ME'E is the pupil magnification » exit pupil diameter/entrance pupil diameter.

Figure 5.2. The paraxial pupil ray and its use in calculation ofm.

Effect of accommodation

The cardinal point positions of the relaxed (zero accommodation) and accommodated versions of schematic eyes can be compared in Figures 5.3, 5.4 and 5.5. Upon accommodation, the principal points move away from the cornea, the nodal points move towards the cornea, and the focal points move towards the cornea.

44 Basic optical structureof Ill,' IIIIII/all e!le

'Exact' schematic eyes

In the 'exact' schematic eyes, an attempt is made to model the optical structure of real

eyes as closely as possible while using spherical surfaces. The minimum requirement of an 'exact' eye is that it must have at least four refracting surfaces, two for the cornea and two for the lens.

Gullstrand number 1 (exact) eye

This schematic eye takes into account the variation of refractive index within the lens (Figure 5.3). It is presented in both relaxed and accommodated versions. It consists of six refracting surfaces; two for the cornea and four for the lens. The lens contains a central nucleus (core) of high refractive index

surrounded by a cortex of lower refractive index. The lens can be regarded as a combination of three lenses. The anterior and posterior lenses are thinner in the centre than at the edge, and may be erroneously considered to have negative power. However, they have positive power because the refractive index of the core lens is higher than that of the cortex.

Gullstrand placed the retina 0.39 mm short of the back focal point F' because he thought

that the positive spherical aberration would lead to the best image plane being slightly in

front of the paraxial image. However, this is arbitrary, because the level of spherical aberration depends upon pupil diameter,

with primary wave spherical aberration depending upon the fourth power of this

l',,,,

.,,,,,

,,,

,

,,

.,

~

Figure 5.3. The Gullstrand number 1 schematic eye.

PP' NN

""j'j""" ---~

"

"

,

"

"

II II

"

PP' NN

Figure 5.4. The Le Grand full theoretical schematic eye.

diameter. Furthermore, the role of spherical aberration may have been greatly exaggerated since real eyes have much less spherical aberration than schematic eyes. We adopt the usual practice of increasing the length of the eye so that the retina coincides with F'.

Le Grand full theoretical eye

The lens of this eye has a constant refractive index, and thus has only two refracting surfaces (Figure 5.4). The eye is presented in both relaxed and accommodated forms.

Simplified schematic eyes

For paraxial calculations, the Gullstrand

number 1 eye and the Le Grand full theoretical eye are more complex than is required for many optical calculations, such as

measurement of retinal image sizes. Simpler

"

"

Relaxed

"

"

PP' NN

Figure 5.5. The Gullstrand-Emsley schematic eye.

eyes are now considered to be adequate. This is because errors that arise in using these simpler models are usually less than the

expected variations between real eyes.

In simplified schematic eyes, the cornea is reduced to a single refracting surface and the lens has two surfaces with a uniform refractive index.

Gullstrand number 2 (simplified) eye as modified by Emsley - the Gullstrand-Emsley eye

Emsley (1952) modified Gullstrand's number

2 eye in order to simplify computation (Figure

5.5). The modifications included altering the

aqueous and vitreous refractive indices to 4/3, altering the lens refractive index to 1.416 for both relaxed and accommodated eyes, thickening the lens and changing the accommodated lens surface radii of curvature

to ±5.00mm.

Paraxialschematiceyes 45

Reduced eyes contain only one refracting surface, which is the cornea. In the exact and simplified eyes already presented, the two principal points and the two nodal points are each separated by values in the range 0.12--0.37 mm. In reduced eyes, the use of a single refracting surface means that its vertex must be at the principal points P(P') and its centre of curvature must be at the nodal points N(N'). To keep a power similar to that

of the more sophisticated eyes, reduced eyes must have shorter axial lengths. As the cornea

has absorbed the power of the lens, the radii of curvature are much smaller than real

values. Since reduced eyes do not have a lens,

they cannot be used to examine the optical consequences of accommodation.

Emsley's reduced eye (1952)

This eye has a corneal radius of curvature of 50/9 mm, a refractive index of 4/3 and a power of 60 D (Figure 5.6).

Le Grand simplified eye

Most of the parameters of this eye are different from those of Le Grand's full schematic eye. The lens is given a zero thickness. The eye has both relaxed and accommodated forms.

Bennett and Rabbetts simplified eye

Bennett and Rabbetts (1988, 1989) modified the relaxed version of the Gullstrand-Emsley eye (see Appendix 3). Rabbetts (1998) introduced forms for accommodation levels of 2.5, 5.0, 7.5 and 10 D. He introduced an 'elderly' version of the eye, which has a lower lens refractive index than do the other forms, and

has a refractive error of 1 D hypermetropia (see Chapter 7).

Reduced schematic eyes

Further simplifications are possible which may give models accurate enough for some calculations, in particular, estimates of retinal image size.

Bennett and Rabbetts (1988, 1989)

This eye has a corneal radius of curvature of 5.6 mm, a refractive index of 1.336 and a power of 60 D.

Variable accommodating eyes

While most of the above models have fixed accommodated forms, none has a variable level of accommodation. As mentioned previously, Blaker (1980) presented a variable accommodating paraxial eye which was later modified to consider aging effects (Blaker,

,,,,,

Figure 5.6. The Emsley reduced schematic eye.

46 Basic optical structureof the humaneye

1991). Navarro et al. (1985) presented a finite accommodating schematic eye which is suitable for easy paraxial calculations because the refractive index of the lens remains uniform, but we leave discussion of this eye

until Chapter 16.

We present here a variable version of Gullstrand's number 1 schematic eye. The eye

was specified at two levels of accommodation (zero and 10.87 D), but we can modify this eye

to have a variable accommodation by assum-

ing that the following individual parameters of this eye vary with accommodation:

anterior chamber depth lens thicknesses

lens cortex anterior curvature lens core anterior curvature

lens core posterior curvature lens cortex posterior curvature.

To simplify the model, we relate the accommodation level A, measured at the corneal vertex, to a parameter x, where

x = 1.052A - 0.00531A2 + 0.000048564A3 (5.12)

and the variable parameters of the eye are related to x by the equations

where the distance unit is millimetres and AQ

is the level of the Gullstrand accommodated eye in dioptres, that is, 10.87013 D.

Equivalent power and positions of cardinal points

tracing, calculate various quantities. For example:

Fa(A) =58.636 + 11.940AIAo 0 N'R'(A) =17.054 - 0.485AIA o mm =11FR - 0.485AIA o mm

where FR is the equivalent power of the relaxed eye. The values of mand the distances of the entrance pupil and the front nodal point

from the anterior corneal vertex (VE and VN)

are plotted as a function of accommodation in Figure 5.7.

Summary of main symbols

A accommodation level at corneal vertex

in dioptres

d surface separations

F equivalent power of the eye

mratio u'ln of paraxial pupil ray angles - this value is a constant for any particular eye at any particular level of

accommodation

MEA pupil magnification, the ratio of

Accommodation (0)

We can now assemble a schematic eye at any level of accommodation and, by paraxial ray-

Figure 5.7. The effect of accommodation on m, VE and VN of a variable accommodating version of the

Gullstrand number 1 schematic eye.